ECON2915 Economic Growth Lecture 2 : The role of capital. The Solow model. Andreas Moxnes University of Oslo Fall 2016 1 / 38
The role of capital Capital is an input in production. Can capital explain why some countries are rich and other poor? ◮ Infrastructure: roads, electricity, water. ◮ Manufacturing: Assembly lines, buildings, computers, robots.. ◮ Agriculture: Tractors, irrigation systems. 2 / 38
Productivity or factor accumulation Recall, income differences Y / L can arise from factor accumulation or productivity. 3 / 38
GDP and capital per worker, 2009 4 / 38
Properties of capital Economic properties of capital: It increases labor productivity: more capital leads to higher output. Not a fixed factor (as opposed to natural resources). A rival good (as opposed to ideas). An excludable good. Yields a return. Depreciates over time. 5 / 38
Economic models over time Before the 1800s, land was the most important input (in addition to labor) Technological change lead to the growing importance of capital. Question: What are the economic properties of land? Agricultural land as a fraction of total wealth in the UK: 6 / 38
Investment Capital accumulation requires investment. In a closed economy, domestic savings = domestic investment. 7 / 38
The production function again Assume Y = F ( K , L ) where Y is output (GDP), K is capital stock, L is labor. Assume Constant returns to scale (CRS). 1 Positive and diminishing marginal product. 2 8 / 38
Assumtion 1 Constant returns to scale means that F ( zK , zL ) = zF ( K , L ) Doubling each input leads to a doubling of output. Alternative terminology: The production function is homogenous of degree n if F ( zK , zL ) = z n F ( K , L ) Homogenous of degree > 1: Increasing returns to scale (IRS). ◮ Increased specialization, volume discounts, etc. Homogenous of degree < 1: Declining returns to scale (DRS). ◮ Too many departments, costly communications etc. Examples? 9 / 38
Assumption 2 Positive marginal product: ′ F K ( K , L ) > 0 Diminishing marginal product: ′′ F KK ( K , L ) < 0 10 / 38
Output per worker We want to express important relationships per capita , e.g. GDP per capita. Start with production function Y = F ( K , L ) L = 1 � K � � K � Y L , L LF ( K , L ) = F = F L , 1 L Define y ≡ Y / L and k ≡ K / L . Then y = F ( k , 1 ) = f ( k ) (intensive form) 11 / 38
Output per worker 12 / 38
A functional form For analytical convenience we use the Cobb-Douglas production function: Y = AK α L 1 − α with 0 < α < 1. Satisfies assumption #1 and #2. What is A ? 13 / 38
Cobb-Douglas: Assumption 1 Constant returns to scale: F ( zK , zL ) = A ( zK ) α ( zL ) 1 − α = z α z 1 − α AK α L 1 − α = zAK α L 1 − α = zF ( K , L ) 14 / 38
Cobb-Douglas: Assumption 2 Recall Y = AK α L 1 − α Positive and diminishing marginal product: K ( K , L ) = A α K α − 1 L 1 − α > 0 ′ F KK = A α ( α − 1 ) K α − 2 L 1 − α < 0 ′′ F Also note that � α � L � 1 − α L = AK α L 1 − α � K Y = A ⇐ ⇒ L L L y = Ak α (intensive form) 15 / 38
What’s α ? Given Cobb-Douglas, α is the share of payments to capital in national income. Why? Assume capital is paid the value of it’s marginal product, r = p × MPK = p α AK α − 1 L 1 − α . Then p α AK α − 1 L 1 − α � � rK K pY = = α pAK α L 1 − α Measured in this way, α is around 1 / 3 according to national accounts. 16 / 38
What’s α ? 17 / 38
What’s α ? (operating profits/(total factor payments) 0.25$ 0.35$ 0.45$ 0.2$ 0.3$ 0.4$ 0.5$ 1970$ 1971$ 1972$ Mainland$ Total$ 1973$ 1974$ 1975$ 1976$ 1977$ 1978$ 1979$ 1980$ 1981$ 1982$ 1983$ 1984$ 1985$ 1986$ 1987$ 1988$ 1989$ 1990$ 1991$ 1992$ 1993$ 1994$ 1995$ 1996$ 1997$ 1998$ 1999$ 2000$ 2001$ 2002$ 2003$ 2004$ 2005$ 2006$ 2007$ 2008$ 2009$ 2010$ 2011$ 2012$ 2013$ 18 / 38 2014$
The Solow model Now we have the prerequisites to analyze the Solow model. The Solow model is fundamental for our understanding of growth and the role of capital. Robert Solow developed the model in 1956, for which he received the Nobel Price in economics in 1987. 19 / 38
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Why a model A mathematical model can clarify complex economic relationships. A simple model can nethertheless yield great insights. Simplifications are necessary because otherwise one loses the big picture. “All theory depends on assumptions which are not quite true. That is what makes it a theory” – Robert Solow (1956, p. 65). 21 / 38
The Solow model - assumptions Closed economy: Y = C + I . Assumptions #1 and #2. For now, A (productivity) and L (population) are constant. Growth (higher y ) can only arise due to capital deepening (higher k ). 22 / 38
The Solow model - capital The change in capital, ∆ K = K t + 1 − K t , is by definition the difference between investment I and depreciation D , ∆ K = I − D . A constant share of output is invested each year, I = γ Y . A constant share of capital depreciates every year, D = δ K . Then ∆ K = γ Y − δ K 23 / 38
Intensive form Express capital accumulation in per capita terms: � K � = ∆ K ∆ k = ∆ L L = γ Y − δ K L = γ y − δ k = γ f ( k ) − δ k . 1st line bc no change in L . 24 / 38
A numerical example Assume that k = 120, f ( k ) = 60 ◮ Capital/income ratio 120 / 60 = 2. Investment is 25% of production and depreciation is 5%. Then ∆ k = γ f ( k ) − δ k = 0 . 25 × 60 − 0 . 05 × 120 = 15 − 6 = 9 . The capital stock per worker increases by 9 units each year. 25 / 38
Steady state We have ∆ k = γ f ( k ) − δ k . A steady state (SS) defined as ∆ k = 0 - investment equals depreciation. Define k ss as the steady state capital stock, γ f ( k ss ) = δ k ss . Define the corresponding SS output level y ss . If k � = k ss , then the capital stock will converge towards k ss . 26 / 38
Steady state k < k ss ⇐ ⇒ γ f ( k ) > δ k ⇐ ⇒ ∆ k > 0 k > k ss ⇐ ⇒ γ f ( k ) < δ k ⇐ ⇒ ∆ k < 0. The steady state is stable . 27 / 38
Higher savings rate More investment/saving -> higher output and capital per capita. 28 / 38
The math With Cobb Douglas, f ( k ) = Ak α . Capital accumulation is then ∆ k = γ Ak α − δ k . The SS is defined by 0 = γ Ak α − δ k δ k = γ Ak α k = γ δ Ak α k 1 − α = γ δ A � γ � 1 / ( 1 − α ) k ss = δ A 29 / 38
The math Output is then y = Ak α y ss = A 1 / ( 1 − α ) � γ � α / ( 1 − α ) δ ⇒ y ss and k ss ↑ . As we saw graphically, γ ↑ = 30 / 38
The speed of convergence Again, ∆ k = γ Ak α − δ k . Divide by k to get the growth rate ˆ k , k = γ Ak α − 1 − δ . Plot the two items on the RHS: ˆ The growth rate of capital/output equals the distance between the curves. → Higher growth far from the SS. − 31 / 38
Predictions Imagine two countries i and j with different savings rates γ i and γ j . A , δ and α are equal in both countries. Both economies are in SS. Then γ is the only source of cross-country income differences. Using the fact that y ss = A 1 / ( 1 − α ) � γ � α / ( 1 − α ) , we get δ � α / ( 1 − α ) y ss � γ i i = y ss γ j j Example: γ i = 0 . 27, γ j = 0 . 03, α = 1 / 3. Then � 1 / 2 y ss � 0 . 27 i = = 3 y ss 0 . 03 j 3x higher GDP/capita in i relative to j . 32 / 38
Theory vs data How much variation in income does the Solow model explain compared to reality? In the model, we only use variation in the investment rate, holding everything else constant. Use average investment rate γ over the 1975-2009 period. α = 1 / 3 for all countries. 33 / 38
Predicted vs actual GDP/capita Much more variation in data than model. Why? 34 / 38
Saving and investment According to model, countries that save more are richer. Caveat: In the model, this is bc investment=saving. In open economy, no equality but investment highly correlated with savings (Chapter 11). 35 / 38
What determines savings? The Solow model predicts that savings determine income. But what determines savings? ◮ Exogenous reasons, i.e. determined outside the model. Culture, geography, demographic variables, government policies (social insurance). ◮ Savings is endogenous, e.g. partly determined by income itself. ⋆ Correlation � = causation, no causal evidence on previous slide. 36 / 38
Example: Endogenous savings Assume that γ = γ 1 if y < y ∗ and γ = γ 2 if y > y ∗ and γ 2 > γ 1 . 37 / 38
Endogenous savings Two equilibria, k ss 1 and k ss 2 . Which one you end up with does not depend on fundamental factors. Instead, depends on the intitial level of income − → s depends on y and y depends on s . Source of poverty traps? 38 / 38
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